SOLUTION: Construct a probability distribution for the sum shown on the faces when two dice, each with 7 faces, are rolled.

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Question 1196729: Construct a probability distribution for the sum shown on the faces when two dice, each with 7 faces, are rolled.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
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This question assumes that you just know on how to do it for the classic 6-faced dice.

Why don't you make it for the 7-faced dice in the same way ?



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Here is what the addition table looks like when we add values from two 7-sided dice
+1234567
12345678
23456789
345678910
4567891011
56789101112
678910111213
7891011121314
For example, if we roll a 1 on a blue die and a 7 on a red die, then we get 1+7 = 8 as shown in the upper right corner of that table.

The possible sums are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

X = sum of the two 7-sided dice
P(X) = probability of that sum showing up

The X values range from 2 to 14 inclusive.
The sum "2" shows up exactly once out of 7*7 = 49 possible outcomes. Therefore P(X) = 1/49 when X = 2
XP(X)
21/49
3
4
5
6
7
8
9
10
11
12
13
14


Then we have the sum "3" show up twice out of 49 possible outcomes. We write 2/49 next to 3 like so
XP(X)
21/49
32/49
4
5
6
7
8
9
10
11
12
13
14


Keep this process going until you have this completed table.
XP(X)
21/49
32/49
43/49
54/49
65/49
76/49
87/49
96/49
105/49
114/49
123/49
132/49
141/49


You could write the table out like this if you prefer
X234567891011121314
P(X)1/492/493/494/495/496/497/496/495/494/493/492/491/49
A pattern to notice is the numerators start to increase {1,2,3,4,5,6,7} when going from X = 2 to X = 8; afterward we have a decreasing set of numerators {6,5,4,3,2,1}
Optionally you could reduce 7/49 to get 1/7, but I find it's better to have all the denominators be the same.

Side note: all of the P(X) values are between 0 and 1. Also, the P(X) values sum to 1.
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